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1: 32.1 Special Notation
§32.1 Special Notation
(For other notation see Notation for the Special Functions.) …
2: 6.1 Special Notation
§6.1 Special Notation
(For other notation see Notation for the Special Functions.) … Unless otherwise noted, primes indicate derivatives with respect to the argument. …
3: 24.1 Special Notation
§24.1 Special Notation
(For other notation see Notation for the Special Functions.) …
Bernoulli Numbers and Polynomials
Among various older notations, the most common one is …
Euler Numbers and Polynomials
4: 1.1 Special Notation
§1.1 Special Notation
(For other notation see Notation for the Special Functions.) …
5: 4.1 Special Notation
§4.1 Special Notation
(For other notation see Notation for the Special Functions.) … The main purpose of the present chapter is to extend these definitions and properties to complex arguments z . …
6: 5.1 Special Notation
§5.1 Special Notation
(For other notation see Notation for the Special Functions.) … The notation Γ ( z ) is due to Legendre. Alternative notations for this function are: Π ( z - 1 ) (Gauss) and ( z - 1 ) ! . Alternative notations for the psi function are: Ψ ( z - 1 ) (Gauss) Jahnke and Emde (1945); Ψ ( z ) Davis (1933); F ( z - 1 ) Pairman (1919). …
7: 12.1 Special Notation
§12.1 Special Notation
(For other notation see Notation for the Special Functions.) … Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. … These notations are due to Miller (1952, 1955). …Whittaker’s notation D ν ( z ) is useful when ν is a nonnegative integer (Hermite polynomial case).
8: 9.1 Special Notation
§9.1 Special Notation
(For other notation see Notation for the Special Functions.)
k nonnegative integer, except in §9.9(iii).
Other notations that have been used are as follows: Ai ( - x ) and Bi ( - x ) for Ai ( x ) and Bi ( x ) (Jeffreys (1928), later changed to Ai ( x ) and Bi ( x ) ); U ( x ) = π Bi ( x ) , V ( x ) = π Ai ( x ) (Fock (1945)); A ( x ) = 3 - 1 / 3 π Ai ( - 3 - 1 / 3 x ) (Szegő (1967, §1.81)); e 0 ( x ) = π Hi ( - x ) , e ~ 0 ( x ) = - π Gi ( - x ) (Tumarkin (1959)).
9: 8.1 Special Notation
§8.1 Special Notation
(For other notation see Notation for the Special Functions.) … Unless otherwise indicated, primes denote derivatives with respect to the argument. … Alternative notations include: Prym’s functions P z ( a ) = γ ( a , z ) , Q z ( a ) = Γ ( a , z ) , Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); ( a , z ) ! = γ ( a + 1 , z ) , [ a , z ] ! = Γ ( a + 1 , z ) , Dingle (1973); B ( a , b , x ) = B x ( a , b ) , I ( a , b , x ) = I x ( a , b ) , Magnus et al. (1966); Si ( a , x ) Si ( 1 - a , x ) , Ci ( a , x ) Ci ( 1 - a , x ) , Luke (1975).
10: 7.1 Special Notation
§7.1 Special Notation
(For other notation see Notation for the Special Functions.) … Unless otherwise noted, primes indicate derivatives with respect to the argument. … Alternative notations are Q ( z ) = 1 2 erfc ( z / 2 ) , P ( z ) = Φ ( z ) = 1 2 erfc ( - z / 2 ) , Erf z = 1 2 π erf z , Erfi z = e z 2 F ( z ) , C 1 ( z ) = C ( 2 / π z ) , S 1 ( z ) = S ( 2 / π z ) , C 2 ( z ) = C ( 2 z / π ) , S 2 ( z ) = S ( 2 z / π ) . The notations P ( z ) , Q ( z ) , and Φ ( z ) are used in mathematical statistics, where these functions are called the normal or Gaussian probability functions. …